Last update: Wednesday, November 2, 11:27am ET.

This is a Stan implementation of Drew Linzer’s dynamic Bayesian election forecasting model, with some tweaks to incorporate national poll data, pollster house effects, correlated priors on state-by-state election results and correlated polling errors.

For more details on the original model:

Linzer, D. 2013. “Dynamic Bayesian Forecasting of Presidential Elections in the States.” Journal of the American Statistical Association. 108(501): 124-134. (link)

The Stan and R files are available here.

1234 polls available since April 01, 2016 (including 928 state polls and 306 national polls).

Electoral College

Note: the model does not account for the specific electoral vote allocation rules in place in Maine and Nebraska.

National Vote

This graph shows Hillary Clinton’s share of the Clinton and Trump national vote, derived from the weighted average of latent state-by-state vote intentions (using the same state weights as in the 2012 presidential election, adjusted for state adult population growth between 2011 and 2015). In the model (described below), national vote intentions are defined as:

\[\pi^{clinton}[t, US] = \sum_{s \in S} \omega_s \cdot \textrm{logit}^{-1} (\mu_a[t] + \mu_b[t, s])\]

The thick line represents the median of posterior distribution of national vote intentions; the light blue area shows the 90% credible interval. The thin blue lines represent 100 draws from the posterior distribution.

From today to November 8, Hillary Clinton’s share of the national vote is predicted to shrink partially towards the fundamentals-based prior (shown with the dotted black line).

Each national poll (raw numbers, unadjusted for pollster house effects) is represented as a dot (darker dots indicate narrower margins of error). On average, Hillary Clinton’s national poll numbers seem to be running slightly below the level that would be consistent with the latent state-by-state vote intentions.

State Vote

The following graphs show vote intention by state (with 100 draws from the posterior distribution represented as thin blue lines):

\[\pi^{clinton}[t,s] = \textrm{logit}^{-1} (\mu_a[t] + \mu_b[t, s])\]

States are sorted by predicted Clinton score on election day.

Current Vote Intentions and Forecast By State

State-by-State Probabilities

Map

Pollster House Effects

Most pro-Clinton polls:

Poll Origin Median P95 P05
Saint Leo University 2.6 1.3 3.9
Public Religion Research Institute 1.8 0.7 2.8
RABA Research 1.7 0.5 2.9
AP 1.6 0.3 2.9
ICITIZEN 1.4 0.2 2.6
McClatchy 1.4 0.0 2.8
GQR 1.3 0.3 2.3
Siena 1.2 -0.3 2.7
Baldwin Wallace University 1.1 -0.6 3.0
University of Delaware 1.1 -0.5 2.8

Most pro-Trump polls:

Poll Origin Median P95 P05
Rasmussen -2.3 -2.9 -1.8
UPI -1.9 -2.5 -1.3
Remington Research Group -1.8 -2.7 -0.9
IBD -1.7 -2.8 -0.7
PPIC -1.7 -3.2 -0.2
Clout Research -1.6 -3.5 0.2
Emerson College Polling Society -1.5 -3.0 0.1
InsideSources -1.5 -3.4 0.3
FOX -1.2 -2.0 -0.5
Dan Jones -1.1 -2.7 0.5

Discrepancy between national polls and weighted average of state polls

Data

The runmodel.R R script downloads state and national polls from the HuffPost Pollster website as .csv files before processing the data.

The model ignores third-party candidates and undecided voters. I restrict each poll’s sample to respondents declaring vote intentions for Clinton or Trump, so that \(N = N^{clinton} + N^{trump}\). (This is problematic for Utah).

When multiple polls are available by the same pollster, at the same date, and for the same state, I pick polls of likely voters rather than registered voters, and polls for which \(N^{clinton} + N^{trump}\) is the smallest (assuming that these are poll questions in which respondents are given the option to choose a third-party candidate, rather than questions in which respondents are only asked to choose between the two leading candidates).

Polls by the same pollster and of the same state with partially overlapping dates are dropped so that only non-overlapping polls are retained, starting from the most recent poll.

To account for the fact that polls can be conducted over several days, I set the poll date to the midpoint between the day the poll started and the day it ended.

Model

The model is in the file state and national polls.stan. It has a backward component, which aggregates poll history to derive unobserved latent vote intentions; and a forward component, which predicts how these unobserved latent vote intentions will evolve until election day. The backward and forward components are linked through priors about vote intention evolution: in each state, latent vote intentions follow a reverse random walk in which vote intentions “start” on election day \(T\) and evolve in random steps (correlated across states) as we go back in time. The starting point of the reverse random walk is the final state of vote intentions, which is assigned a reasonable prior, based on the Time-for-change, fundamentals-based electoral prediction model. The model reconciles the history of state and national polls with prior beliefs about final election results and about how vote intentions evolve.

Backward Component: Poll Aggregation

For each poll \(i\), the number of respondents declaring they intended to vote for Hillary Clinton \(N^{clinton}_i\) is drawn from a binomial distribution:

\[ N^{clinton}_i \sim \textrm{Binomial}(N_i, \pi^{clinton}_i) \]

where \(N_i\) is poll sample size, and \(\pi^{clinton}_i\) is share of the Clinton vote for this poll.

The model treats national and state polls differently.

State polls

If poll \(i\) is a state poll, I use a day/state/pollster multilevel model:

\[\textrm{logit} (\pi^{clinton}_i) = \mu_a[t_i] + \mu_b[t_i, s_i] + \mu_c[p_i] + u_i + e[s_i]\]

What this model does is simply to decompose the log-odds of reported vote intentions towards Hillary Clinton \(\pi^{clinton}_i\) into a national component, shared across all states (\(\mu_a\)), a state-specific component (\(\mu_b\)), a pollster house effect (\(\mu_c\)), a poll-specific measurement noise term (\(u\)), and a polling error term (\(e\)) shared across all polls of the state (the higher \(e\), the more polls overestimate Hillary Clinton’s true score).

On the day of the last available poll \(t_{last}\), the national component \(\mu_a[t_{last}]\) is set to zero, so that the predicted share of the Clinton vote in state \(s\) (net of pollster house effects and measurement noise) after that date and until election day \(T\) is:

\[\pi^{clinton}_{ts} = \textrm{logit}^{-1} (\mu_b[t, s])\]

To reduce the number of parameters, the model only takes weekly values for \(\mu_b\), so that:

\[\mu_b[t, s] = \mu_b^{weekly}[w_t, s]\]

where \(w_t\) is the week of day \(t\).

National polls

If poll \(i\) is a national poll, I use the same multilevel approach (with random intercepts for pollster house effects \(\mu_c\)) but I add a little tweak: the share of the Clinton vote in a national poll should also reflect the weighted average of state-by-state scores at the time of the poll. I model the share of vote intentions in national polls in the following way:

\[\textrm{logit} (\pi^{clinton}_i) = \textrm{logit}\left( \sum_{s \in \{1 \dots S\}} \omega_s \cdot \textrm{logit}^{-1} (\mu_a[t_i] + \mu_b^{weekly}[w_{t_i}, s] + e[s]) \right) + \alpha + \mu_c[p_i] + u_i\]

where \(\omega_s\) represents the share of state \(s\) in the total votes of the set of polled states \(1 \dots S\) (based on 2012 turnout numbers adjusted for adult population growth in each state between 2011 and 2015). The \(\alpha\) parameter corrects for possible discrepancies between national polls and the weighted average of state polls. Possible sources of discrepancies may include:

  • the fact that when polls are not available for all states, polled states can be on average more blue or more red than the country as a whole (not a problem since the first 50-state Washington Post/SurveyMonkey poll in early September);
  • changes in state weights since 2012;
  • any possible (time-invariant) bias in national polls relative to state polls.

The idea is that while national poll levels may be off and generally not very indicative of the state of the race, national poll changes may contain valuable information to update \(\mu_a\) and (to a lesser extent) \(\mu_b\) parameters.

How vote intentions evolve

In order to smooth out vote intentions by state and obtain latent vote intentions at dates in which no polls were conducted, I use 2 reverse random walk priors for \(\mu_a\) and \(\mu_b^{weekly}\) from \(t_{last}\) to April 1:

\[\mu_b^{weekly}[w_t-1, s] \sim \textrm{Normal}(\mu_b^{weekly}[w_t, s], \sigma_b \cdot \sqrt{7})\]

\[\mu_a[t-1] \sim \textrm{Normal}(\mu_a[t], \sigma_a)\]

Both \(\sigma_a\) and \(\sigma_b\) are given uniform priors between 0 and 0.05.

Their posterior marginal distributions are shown below. The median day-to-day total standard deviation of vote intentions is about 0.4%. The model seems to find that most of the changes in latent vote intentions are attributable to national swings rather than state-specific swings (national swings account on average for about 90% of the total day-to-day variance).

Forward Component: Vote Intention Forecast

Final outcome

I use a multivariate normal distribution for the prior of the final outcome. Its mean is based on the Time-for-Change model – which predicts that Hillary Clinton should receive 48.6% of the national vote (based on Q2 GDP figures, the current President’s approval rating and number of terms). The prior expects state final scores to remain on average centered around \(48.6\% + \delta_s\), where \(\delta_s\) is the excess Obama performance relative to the national vote in 2012 in state \(s\).

\[\mu_b[T, 1 \dots S] \sim \textrm{Multivariate Normal}(\textrm{logit} (0.486 + \delta_{1 \dots S}), \mathbf{\Sigma})\]

For the covariance matrix \(\mathbf{\Sigma}\), I set the variance to 0.05 and the covariance to 0.025 for all states and pairs of states – which corresponds to a correlation coefficient of 0.5 across states.

  • This prior is relatively imprecise as to the expected final scores in any given state; for example, in a state like Virginia, which Obama won by 52% in 2012 (a score identical to his national score), Hillary Clinton is expected to get 48.6% of the vote, with a 95% certainty that her score will not fall below 38% or exceed 59%.

  • State scores are also expected to be correlated with each other. For example, according to the prior (before looking at polling data), there is only a 3.4% chance that Hillary Clinton will perform worse in Virginia than in Texas. If the priors were independent, this unlikely event could happen with a 10% probability.

The covariance matrix implies that the correlation between the 2012 state scores and 2016 state priors is expected to be about 0.94 (as opposed to 0.89 if covariances were set to zero). The simulated distribution of correlations between state priors and 2012 scores is in line with observed correlations of state scores with previous election results since 1988 [http://election.princeton.edu/2016/06/02/the-realignment-myth/].

To put it differently, the model does not have a very precise prior about final scores, but it does assume that most of this uncertainty is attributable to national-level swings in vote intentions.

How vote intentions evolve

From election day to the date of the latest available poll \(t_{last}\), vote intentions by state “start” at \(\mu_b[T,s]\) and follow a random walk with correlated steps across states:

\[\mu_b^{weekly}[w_t-1, 1 \dots S] \sim \textrm{Multivariate Normal}(\mu_b^{weekly}[w_t, 1 \dots S], \mathbf{\Sigma_b^{walk}})\]

I set \(\mathbf{\Sigma_b^{walk}}\) so that all variances equal \(0.015^2 \times 7\) and all covariances equal 0.00118 (\(\rho =\) 0.75). This implies a 0.4% standard deviation in daily vote intentions changes in a state where Hillary Clinton’s score is close to 50%. To put it differently, the prior is 95% confident that Hillary Clinton’s score in any given state where she is currently polling around 50% should not move up or down by more than 1.8% over the remaining 6 days until the election.

Poll house effects

Each pollster \(p\) can be biased towards Clinton or Trump:

\[\mu_c[p] \sim \textrm{Normal}(0, \sigma_c)\]

\[\sigma_c \sim \textrm{Uniform}(0, 0.2)\]

Discrepancy between national polls and the average of state polls

I give the \(\alpha\) parameter a prior centered around the observed distance of polled state voters from the national vote in 2012 (this was useful until early September, when lots of solid red states had still not been polled and the average polled state voter was more pro-Clinton than the average US voter.):

\[\bar{\delta_S} = \sum_{s \in \{1 \dots S\}} \omega_s \cdot \pi^{obama'12}_s - \pi^{obama'12}\]

\[\alpha \sim \textrm{Normal}(\textrm{logit} (\bar{\delta_S}), 0.2)\]

Measurement noise

The measurement noise term \(u_i\) is normally distributed around zero, with standard error \(\sigma_u^{national}\) for national polls, and \(\sigma_u^{state}\) for state polls. I give both standard errors a uniform distribution between 0 and 0.10.

\[\sigma_u^{national} \sim \textrm{Uniform}(0, 0.1)\] \[\sigma_u^{state} \sim \textrm{Uniform}(0, 0.1)\]

Polling error

To account for the possibility that polls might be off on average, even after adjusting for pollster house effects, the model includes a polling error term shared by all polls of the same state \(e[s]\). For example, the presence of an unexpectedly large share of Trump voters (undetected by the polls) in a given state would translate into large positive \(e\) values for that state. This polling error will remain unknown until election day; however it can be included in the form of an unidentified random parameter in the likelihood of the model, that increases the uncertainty in the posterior distribution of \(\mu_a\) and \(\mu_b\).

Because I expect polling errors to be correlated across states, I use a multivariate normal distribution:

\[e \sim \textrm{Multivariate Normal}(0, \mathbf{\Sigma_e})\]

To construct \(\mathbf{\Sigma_e}\), I set the variance to \(0.04^2\) and the covariance to 0.00175; this corresponds to a standard deviation of about 1 percentage point for a state in which Clinton’s score is close to 50% (or a 95% certainty that polls are not off by more than 2 percentage points either way); and a 0.7 correlation of polling errors across states.

Recently added polls

Entry Date Source State % Clinton / (Clinton + Trump) % Trump / (Clinton + Trump) N (Clinton + Trump)
2016-11-02 ABC – 50.0 50.0 1087
2016-11-02 IBD – 50.0 50.0 759
2016-11-02 Rasmussen – 50.0 50.0 1320
2016-11-02 YouGov – 51.7 48.3 1097
2016-11-02 SurveyMonkey AK 44.6 55.4 222
2016-11-02 SurveyMonkey AL 41.6 58.4 448
2016-11-02 SurveyMonkey AR 42.2 57.8 428
2016-11-02 SurveyMonkey AZ 49.4 50.6 1122
2016-11-02 SurveyMonkey CA 64.7 35.3 1941
2016-11-02 SurveyMonkey CO 51.8 48.2 1164
2016-11-02 SurveyMonkey CT 57.3 42.7 548
2016-11-02 SurveyMonkey DE 56.8 43.2 363
2016-11-02 SurveyMonkey FL 51.6 48.4 2471
2016-11-02 TargetSmart FL 54.5 45.5 632
2016-11-02 SurveyMonkey GA 49.4 50.6 2383
2016-11-02 SurveyMonkey HI 63.0 37.0 347
2016-11-02 SurveyMonkey IA 46.4 53.6 728
2016-11-02 SurveyMonkey ID 40.5 59.5 311
2016-11-02 SurveyMonkey IL 59.8 40.2 793
2016-11-02 SurveyMonkey IN 41.6 58.4 568
2016-11-02 SurveyMonkey KS 43.5 56.5 989
2016-11-02 SurveyUSA KS 43.7 56.3 543
2016-11-02 SurveyMonkey KY 37.4 62.6 454
2016-11-02 Lucid LA 48.2 51.8 510
2016-11-02 SurveyMonkey LA 42.0 58.0 414
2016-11-02 SurveyMonkey MA 66.7 33.3 693
2016-11-02 SurveyMonkey MD 70.0 30.0 626
2016-11-02 SurveyMonkey ME 57.0 43.0 368
2016-11-02 SurveyMonkey MI 50.6 49.4 1513
2016-11-02 SurveyMonkey MN 56.0 44.0 676
2016-11-02 SurveyMonkey MO 45.3 54.7 558
2016-11-02 SurveyMonkey MS 46.6 53.4 548
2016-11-02 SurveyMonkey MT 40.5 59.5 323
2016-11-02 SurveyMonkey NC 53.8 46.2 1471
2016-11-02 SurveyMonkey ND 37.2 62.8 218
2016-11-02 SurveyMonkey NE 38.8 61.2 374
2016-11-02 SurveyMonkey NH 58.3 41.7 533
2016-11-02 SurveyMonkey NJ 60.4 39.6 654
2016-11-02 SurveyMonkey NM 55.3 44.7 684
2016-11-02 SurveyMonkey NV 49.4 50.6 845
2016-11-02 SurveyMonkey NY 63.3 36.7 1480
2016-11-02 SurveyMonkey OH 48.3 51.7 1380
2016-11-02 SurveyMonkey OK 36.8 63.2 452
2016-11-02 SurveyMonkey OR 58.6 41.4 704
2016-11-02 SurveyMonkey PA 53.3 46.7 1870
2016-11-02 SurveyMonkey RI 57.6 42.4 347
2016-11-02 SurveyMonkey SC 49.4 50.6 1413
2016-11-02 SurveyMonkey SD 34.2 65.8 222
2016-11-02 SurveyMonkey TN 44.0 56.0 639
2016-11-02 SurveyMonkey TX 46.6 53.4 1733
2016-11-02 SurveyMonkey UT 46.9 53.1 676
2016-11-02 SurveyMonkey VA 55.7 44.3 1698
2016-11-02 Winthrop University VA 53.0 47.0 591
2016-11-02 SurveyMonkey VT 67.5 32.5 342
2016-11-02 SurveyMonkey WA 61.4 38.6 614
2016-11-02 SurveyMonkey WI 51.2 48.8 949
2016-11-02 SurveyMonkey WV 34.5 65.5 267
2016-11-01 Lucid – 51.2 48.8 710
2016-11-01 UPI – 50.5 49.5 1260
2016-11-01 UPI AK 42.7 57.3 292
2016-11-01 UPI AL 38.9 61.1 332
2016-11-01 UPI AR 38.9 61.1 308
2016-11-01 Data Orbital AZ 47.7 52.3 473
2016-11-01 UPI AZ 45.7 54.3 351
2016-11-01 SurveyUSA CA 61.5 38.5 680
2016-11-01 UPI CA 60.4 39.6 462
2016-11-01 UPI CO 52.1 47.9 328
2016-11-01 UPI CT 55.7 44.3 316
2016-11-01 UPI DE 56.7 43.3 299
2016-11-01 UPI FL 50.5 49.5 388
2016-11-01 UPI GA 47.4 52.6 347
2016-11-01 UPI HI 66.0 34.0 301
2016-11-01 UPI IA 50.0 50.0 316
2016-11-01 UPI ID 35.8 64.2 302
2016-11-01 Loras College IL 57.0 43.0 474
2016-11-01 UPI IL 57.7 42.3 375
2016-11-01 UPI IN 43.8 56.2 319
2016-11-01 UPI KS 41.1 58.9 312
2016-11-01 Cygnal KY 36.4 63.6 714
2016-11-01 UPI KY 38.5 61.5 311
2016-11-01 UPI LA 41.7 58.3 303
2016-11-01 UPI MA 59.8 40.2 315
2016-11-01 UPI MD 60.4 39.6 317
2016-11-01 MPRC ME 53.2 46.8 641
2016-11-01 UPI ME 55.7 44.3 303
2016-11-01 UPI MI 52.6 47.4 327
2016-11-01 UPI MN 51.5 48.5 307
2016-11-01 Monmouth University MO 42.2 57.8 364
2016-11-01 UPI MO 45.4 54.6 363
2016-11-01 UPI MS 43.3 56.7 304
2016-11-01 UPI MT 42.7 57.3 326
2016-11-01 Elon NC 50.6 49.4 589
2016-11-01 SurveyUSA NC 46.3 53.7 626
2016-11-01 UPI NC 50.0 50.0 337
2016-11-01 UPI ND 39.4 60.6 288
2016-11-01 UPI NE 39.6 60.4 321
2016-11-01 UNH NH 54.1 45.9 525
2016-11-01 UPI NH 51.5 48.5 298
2016-11-01 UPI NJ 58.8 41.2 342
2016-11-01 UPI NM 53.2 46.8 303
2016-11-01 UPI NV 51.1 48.9 295
2016-11-01 UPI NY 60.4 39.6 387
2016-11-01 UPI OH 51.5 48.5 329
2016-11-01 UPI OK 34.4 65.6 308
2016-11-01 UPI OR 53.8 46.2 291
2016-11-01 Franklin and Marshall College PA 56.3 43.7 567
2016-11-01 UPI PA 52.1 47.9 351
2016-11-01 UPI RI 58.8 41.2 299
2016-11-01 UPI SC 43.8 56.2 312
2016-11-01 UPI SD 41.7 58.3 299
2016-11-01 UPI TN 40.0 60.0 322
2016-11-01 UPI TX 42.7 57.3 412
2016-11-01 UPI UT 30.5 69.5 299
2016-11-01 UPI VA 52.1 47.9 324
2016-11-01 Washington Post VA 53.3 46.7 922
2016-11-01 UPI VT 64.9 35.1 295
2016-11-01 UPI WA 55.2 44.8 314
2016-11-01 UPI WI 52.6 47.4 354
2016-11-01 UPI WV 37.9 62.1 295
2016-11-01 UPI WY 31.6 68.4 295

Convergence checks

With 4 chains and 2000 iterations (the first 1000 iterations of each chain are discarded), the model runs in about 10 minutes on my 4-core Intel i7 MacBookPro.

##  [1] "Inference for Stan model: state and national polls."                          
##  [2] "4 chains, each with iter=2000; warmup=1000; thin=1; "                         
##  [3] "post-warmup draws per chain=1000, total post-warmup draws=4000."              
##  [4] ""                                                                             
##  [5] "                   mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat"
##  [6] "alpha             -0.01       0 0.01 -0.03 -0.02 -0.01 -0.01  0.00  2173    1"
##  [7] "sigma_c            0.05       0 0.01  0.04  0.05  0.05  0.05  0.06  1160    1"
##  [8] "sigma_u_state      0.05       0 0.00  0.04  0.05  0.05  0.06  0.06  1180    1"
##  [9] "sigma_u_national   0.01       0 0.01  0.00  0.01  0.01  0.02  0.03   743    1"
## [10] "sigma_walk_a_past  0.02       0 0.00  0.01  0.02  0.02  0.02  0.02  1183    1"
## [11] "sigma_walk_b_past  0.01       0 0.00  0.00  0.00  0.01  0.01  0.01   945    1"
## [12] "mu_b[33,2]        -0.22       0 0.07 -0.37 -0.27 -0.22 -0.17 -0.08  3549    1"
## [13] "mu_b[33,3]        -0.40       0 0.07 -0.53 -0.45 -0.40 -0.36 -0.27  3714    1"
## [14] "mu_b[33,4]        -0.39       0 0.07 -0.53 -0.44 -0.39 -0.34 -0.25  3735    1"
## [15] "mu_b[33,5]        -0.07       0 0.06 -0.19 -0.11 -0.07 -0.03  0.05  3538    1"
## [16] "mu_b[33,6]         0.60       0 0.06  0.48  0.56  0.60  0.65  0.73  3421    1"
## [17] "mu_b[33,7]         0.08       0 0.06 -0.05  0.04  0.08  0.12  0.20  3216    1"
## [18] "mu_b[33,8]         0.28       0 0.07  0.15  0.24  0.28  0.33  0.42  3547    1"
## [19] "mu_b[33,9]         0.35       0 0.07  0.21  0.30  0.35  0.40  0.50  3778    1"
## [20] "mu_b[33,10]        0.03       0 0.06 -0.09 -0.01  0.03  0.07  0.15  3517    1"
## [21] "mu_b[33,11]       -0.09       0 0.06 -0.21 -0.13 -0.09 -0.05  0.04  3493    1"
## [22] "mu_b[33,12]        0.61       0 0.08  0.45  0.56  0.61  0.67  0.77  4000    1"
## [23] "mu_b[33,13]       -0.06       0 0.07 -0.19 -0.10 -0.06 -0.01  0.07  3446    1"
## [24] "mu_b[33,14]       -0.55       0 0.07 -0.69 -0.60 -0.55 -0.50 -0.41  3648    1"
## [25] "mu_b[33,15]        0.38       0 0.07  0.25  0.34  0.38  0.43  0.51  3463    1"
## [26] "mu_b[33,16]       -0.31       0 0.06 -0.43 -0.35 -0.31 -0.26 -0.18  3697    1"
## [27] "mu_b[33,17]       -0.31       0 0.07 -0.44 -0.36 -0.31 -0.27 -0.18  3424    1"
## [28] "mu_b[33,18]       -0.51       0 0.07 -0.64 -0.55 -0.51 -0.46 -0.37  3706    1"
## [29] "mu_b[33,19]       -0.35       0 0.07 -0.48 -0.40 -0.35 -0.31 -0.22  3547    1"
## [30] "mu_b[33,20]        0.60       0 0.07  0.47  0.55  0.60  0.64  0.73  3347    1"
## [31] "mu_b[33,21]        0.66       0 0.07  0.52  0.61  0.66  0.70  0.79  3351    1"
## [32] "mu_b[33,22]        0.18       0 0.07  0.05  0.14  0.18  0.23  0.31  3386    1"
## [33] "mu_b[33,23]        0.10       0 0.06 -0.03  0.06  0.10  0.14  0.22  3114    1"
## [34] "mu_b[33,24]        0.16       0 0.07  0.03  0.12  0.16  0.21  0.30  3350    1"
## [35] "mu_b[33,25]       -0.22       0 0.06 -0.35 -0.27 -0.22 -0.18 -0.09  3656    1"
## [36] "mu_b[33,26]       -0.23       0 0.07 -0.37 -0.28 -0.23 -0.18 -0.09  3810    1"
## [37] "mu_b[33,27]       -0.37       0 0.08 -0.52 -0.42 -0.37 -0.32 -0.22  3604    1"
## [38] "mu_b[33,28]        0.02       0 0.06 -0.10 -0.02  0.02  0.06  0.14  3670    1"
## [39] "mu_b[33,29]       -0.47       0 0.08 -0.64 -0.52 -0.47 -0.42 -0.31  4000    1"
## [40] "mu_b[33,30]       -0.41       0 0.07 -0.56 -0.46 -0.41 -0.36 -0.27  4000    1"
## [41] "mu_b[33,31]        0.14       0 0.06  0.01  0.09  0.14  0.18  0.26  3601    1"
## [42] "mu_b[33,32]        0.35       0 0.07  0.21  0.31  0.35  0.39  0.48  3566    1"
## [43] "mu_b[33,33]        0.20       0 0.07  0.07  0.15  0.20  0.25  0.34  3677    1"
## [44] "mu_b[33,34]        0.01       0 0.06 -0.12 -0.03  0.01  0.05  0.13  3371    1"
## [45] "mu_b[33,35]        0.51       0 0.07  0.38  0.47  0.51  0.56  0.64  2820    1"
## [46] "mu_b[33,36]       -0.03       0 0.06 -0.15 -0.08 -0.03  0.01  0.09  2694    1"
## [47] "mu_b[33,37]       -0.56       0 0.07 -0.69 -0.60 -0.55 -0.51 -0.42  3525    1"
## [48] "mu_b[33,38]        0.24       0 0.07  0.11  0.20  0.24  0.29  0.37  3428    1"
## [49] "mu_b[33,39]        0.11       0 0.06 -0.01  0.07  0.11  0.15  0.23  3133    1"
## [50] "mu_b[33,40]        0.33       0 0.08  0.18  0.28  0.33  0.38  0.48  4000    1"
## [51] "mu_b[33,41]       -0.12       0 0.07 -0.25 -0.16 -0.12 -0.07  0.01  3180    1"
## [52] "mu_b[33,42]       -0.40       0 0.08 -0.56 -0.45 -0.40 -0.35 -0.24  4000    1"
## [53] "mu_b[33,43]       -0.36       0 0.07 -0.50 -0.41 -0.36 -0.32 -0.23  3488    1"
## [54] "mu_b[33,44]       -0.21       0 0.06 -0.34 -0.25 -0.21 -0.16 -0.09  3590    1"
## [55] "mu_b[33,45]       -0.36       0 0.07 -0.49 -0.41 -0.36 -0.32 -0.24  3667    1"
## [56] "mu_b[33,46]        0.16       0 0.06  0.04  0.12  0.17  0.21  0.28  3547    1"
## [57] "mu_b[33,47]        0.76       0 0.08  0.61  0.71  0.76  0.81  0.91  4000    1"
## [58] "mu_b[33,48]        0.32       0 0.07  0.18  0.27  0.31  0.36  0.45  3377    1"
## [59] "mu_b[33,49]        0.11       0 0.06 -0.02  0.07  0.11  0.15  0.23  3420    1"
## [60] "mu_b[33,50]       -0.57       0 0.07 -0.72 -0.62 -0.57 -0.53 -0.43  4000    1"
## [61] "mu_b[33,51]       -0.92       0 0.08 -1.08 -0.97 -0.92 -0.86 -0.76  4000    1"
## [62] ""                                                                             
## [63] "Samples were drawn using NUTS(diag_e) at Wed Nov  2 16:26:11 2016."           
## [64] "For each parameter, n_eff is a crude measure of effective sample size,"       
## [65] "and Rhat is the potential scale reduction factor on split chains (at "        
## [66] "convergence, Rhat=1)."